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Equations of \(\overline{M}_{0,n}\)

  • Leonid Monin
  • Julie RanaEmail author
Chapter
Part of the Fields Institute Communications book series (FIC, volume 80)

Abstract

We study the moduli space \(\overline{M}_{0,n}\) of genus 0 curves with n marked points. Following Keel and Tevelev, we give explicit polynomials in the Cox ring of \(\mathbb{P}^{1} \times \mathbb{P}^{2} \times \cdots \times \mathbb{P}^{n-3}\) that, conjecturally, determine \(\overline{M}_{0,n}\) as a subscheme. Using Macaulay2, we prove that these equations generate the ideal for 5 ≤ n ≤ 8. For n ≤ 6, we also give a cohomological proof that these polynomials realize \(\overline{M}_{0,n}\) as a subvariety of \(\mathbb{P}^{(n-2)!-1}\) embedded by the complete log canonical linear system.

MSC 2010 codes:

14H10 (primary) 13D02 (secondary) 

Notes

Acknowledgements

This article was initiated during the Apprenticeship Weeks (22 August–2 September 2016), led by Bernd Sturmfels, as part of the Combinatorial Algebraic Geometry Semester at the Fields Institute. We thank Bernd Sturmfels for providing inspiration and feedback on multiple drafts. We are also grateful to Christine Berkesch Zamaere, Renzo Cavalieri, Diane Maclagan, Steffen Marcus, Vic Reiner, and Jenia Tevelev for many helpful discussions. The second author was partially supported by a scholarship from the Clay Math Institute.

References

  1. 1.
    Renzo Cavalieri: Moduli spaces of pointed rational curves, Graduate Summer School, 2016 Combinatorial Algebraic Geometry Program, Fields Institute, Toronto, www.math.colostate.edu/~renzo/teaching/Moduli16/Fields.pdf.
  2. 2.
    Melody Chan: Lectures on tropical curves and their moduli spaces, arXiv:1606.02778 [math.AG].Google Scholar
  3. 3.
    Ana-Maria Castravet and Jenia Tevelev: \(\overline{M}_{0,n}\) is not a Mori dream space, Duke Math. J. 164 (2015) 1641–1667.Google Scholar
  4. 4.
    Pierre Deligne and David Mumford: The irreducibility of the space of curves of given genus, Inst. Hautes Études Sci. Publ. Math. 36 (1969) 75–109.Google Scholar
  5. 5.
    Maksym Fedorchuk and David Ishii Smyth: Alternate compactifications of moduli spaces of curves, in Handbook of moduli I, 331–413, Adv. Lect. Math. 24, Int. Press, Somerville, MA, 2016.Google Scholar
  6. 6.
    José González and Kalle Karu: Some non-finitely generated Cox rings, Compos. Math. 152 (2016) 984–996.Google Scholar
  7. 7.
    Angela Gibney, Sean Keel, and Ian Morrison: Towards the ample cone of \(\overline{M}_{0,n}\), J. Amer. Math. Soc. 15 (2002) 273–294.Google Scholar
  8. 8.
    Angela Gibney and Diane Maclagan: Equations for Chow and Hilbert quotients, Algebra Number Theory 4 (2010) 855–885.Google Scholar
  9. 9.
    Yi Hu and Sean Keel: Mori dream spaces and GIT, Michigan Math. J. 48 (2000) 331–348.Google Scholar
  10. 10.
    Joe Harris and David Mumford: On the Kodaira dimension of the moduli space of curves, Invent. Math. 67 (1982) 23–88.Google Scholar
  11. 11.
    Joe Harris and Ian Morrison: Moduli of curves, Graduate Texts in Mathematics 187, Springer-Verlag, New York, 1998.Google Scholar
  12. 12.
    Benjamin Howard, John Millson, Andrew Snowden, and Ravi Vakil: The equations for the moduli space of n points on the line, Duke Math. J. 146 (2009) 175–226.Google Scholar
  13. 13.
    Mikhail M. Kapranov: Chow quotients of Grassmannians I, in I.M.Gel’fand Seminar, 29–110, Adv. Soviet Math. 16, American. Mathematical. Society, Providence, RI, 1993.Google Scholar
  14. 14.
    _________ : Veronese curves and Grothendieck-Knudsen moduli space \(\overline{M}_{0,n}\), J. Algebraic Geom. 2 (1993) 239–262.Google Scholar
  15. 15.
    Sean Keel: Intersection theory of moduli space of stable n-pointed curves of genus zero, Trans. Amer. Math. Soc. 330 (1992) 545–574.Google Scholar
  16. 16.
    Sean Keel and Jenia Tevelev: Equations for \(\overline{M}_{0,n}\), Internat. J. Math. 20 (2009) 1159–1184.Google Scholar
  17. 17.
    Andrei Losev and Yuri Manin: New moduli spaces of pointed curves and pencils of flat connections, Michigan Math. J. 48 (2000) 443–472.Google Scholar
  18. 18.
    Diane Maclagan and Bernd Sturmfels: Introduction to Tropical Geometry, Graduate Studies in Mathematics 161, American Mathematical Society, RI, 2015.Google Scholar
  19. 19.
    Ranjan Roy: Binomial identities and hypergeometric series, Amer. Math. Monthly 94 (1987) 36–46.Google Scholar
  20. 20.
    Bernd Sturmfels: Fitness, apprenticeship, and polynomials, in Combinatorial Algebraic Geometry, 1–19, Fields Inst. Commun. 80, Fields Inst. Res. Math. Sci., 2017.Google Scholar

Copyright information

© Springer Science+Business Media LLC 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of TorontoTorontoCanada
  2. 2.School of MathematicsUniversity of MinnesotaMinneapolisUSA

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